Question

Ken wants to build a table and put a border around it. The table and border must have an area of 3,276 square feet. The table is 36 inches wide and 72 inches long without the border. Which quadratic equation can be used to determine the thickness of the border, x?

Answer 1

`4\cdot x^(2) + 18\cdot x -3258\,ft^(2) = 0`,
`x \approx 26.378\,ft`

Explanation:

The area of the table and border can be computed by using the following second order-polynomial:

`3276\,ft^(2) = (2\cdot x + (36)/(12)\,ft )\cdot (2\cdot x + (72)/(12)\,ft )`

`3276\,ft^(2) = (2\cdot x + 3\,ft)\cdot (2\cdot x + 6\,ft)`

The thickness of the border can be determined by expanding and simplyfiying the algebraic expression:

`3276\,ft^(2) = (2\cdot x)^(2) + 9\cdot (2\cdot x) + 18\,ft^(2)`

`4\cdot x^(2) + 18\cdot x -3258\,ft^(2) = 0`

The roots of the polynomial are:

`x_(1) \approx 26.378\,ft` and
`x_(2) \approx -30.878\,ft`

Knowing that length is a positive unit, the first root is the only solution that is reasonable:

`x \approx 26.378\,ft`

Answer 2

Let t be the thickness of the border. Adding the border to all side of the table will increase the length and the width by 2t. The area of a rectangle is the product of the length and width. With the addition of the thickness of the border, the new quadratic equation becomes,

(36 + 2t)(72 + 2t) = 3,276.